Statistical Inference Serik Sagitov, Chalmers University of Technology and Gothenburg University Abstract This text is a compendium for the undergraduate course on course MVE155 "Statistical Inference" worth of 7.5 hp, which is a second course in mathematical statistics suitable for students with different backgrounds. A main prerequisite is an introductory course in probability and statistics. The course gives a deeper understanding of some traditional topics in mathematical statistics such as methods based on likelihood, topics in descriptive statistics and data analysis with special attention to graphical displays, aspects of experimental design. The course introduces a number of standard tools of statistical inference including bootstrap, parametric and non-parametric testing, analysis of variance, introduction to Bayesian inference, chi-squared tests, multiple regression. Recommended textbook: John Rice, Mathematical statistics and data analysis, 3rd edition. The com- pendium includes a collection of solved exercises many of which are the end-of-chapter exercises from the Rice's book. Do not read a solution before you tried to solve an exercise on your own. Please send your corrections to [email protected]. (I gratefully acknowledge multiple corrections kindly proposed by the students of the year 2019.) Last updated: June 10, 2019 Contents Abstract 1 1 Introduction 2 1.1 List of course topics: part I . 3 1.2 List of course topics: part II . 3 2 Random sampling 5 2.1 Point estimation . 5 2.2 Sample mean and sample variance . 6 2.3 Approximate confidence intervals . 7 2.4 Stratified random sampling . 7 2.5 Exercises . 8 3 Parameter estimation 10 3.1 Method of moments . 10 3.2 Maximum likelihood estimation . 11 3.3 Sufficiency . 12 3.4 Large sample properties of the maximum likelihood estimates . 12 3.5 Gamma distribution . 13 3.6 Exact confidence intervals . 15 3.7 Exercises . 16 4 Hypothesis testing 19 4.1 Statistical significance . 19 4.2 Large-sample test for the proportion . 19 4.3 Small-sample test for the proportion . 21 4.4 Two tests for the mean . 21 4.5 Likelihood ratio test . 21 4.6 Pearson's chi-squared test . 22 4.7 Example: sex ratio . 23 4.8 Exercises . 23 1 5 Bayesian inference 27 5.1 Conjugate priors . 27 5.2 Bayesian estimation . 29 5.3 Credibility interval . 30 5.4 Bayesian hypotheses testing . 31 5.5 Exercises . 32 6 Summarising data 34 6.1 Empirical probability distribution . 34 6.2 Density estimation . 36 6.3 Quantiles and QQ-plots . 36 6.4 Testing normality . 37 6.5 Measures of location . 38 6.6 Measures of dispersion . 39 6.7 Exercises . 39 7 Comparing two samples 41 7.1 Two independent samples: comparing population means . 41 7.2 Two independent samples: comparing population proportions . 44 7.3 Paired samples . 45 7.4 External and confounding factors . 47 7.5 Exercises . 48 8 Analysis of variance 51 8.1 One-way layout . 51 8.2 Simultaneous confidence interval . 53 8.3 Kruskal-Wallis test . 54 8.4 Two-way layout . 55 8.5 Example: iron retention . 55 8.6 Randomised block design . 57 8.7 Friedman test . 59 8.8 Exercises . 59 9 Categorical data analysis 61 9.1 Chi-squared test of homogeneity . 62 9.2 Chi-squared test of independence . 62 9.3 Matched-pairs designs . 63 9.4 Odds ratios . 65 9.5 Exercises . 65 10 Multiple regression 68 10.1 Simple linear regression model . 68 10.2 Residuals . 69 10.3 Confidence intervals and hypothesis testing . 70 10.4 Intervals for individual observations . 71 10.5 Linear regression and ANOVA . 72 10.6 Multiple linear regression . 72 10.7 Exercises . 75 11 Overview of basic probability theory 77 11.1 Probability rules . 77 11.2 Random variables . 77 11.3 Important parametric distributions . 78 11.4 Joint distributions . 79 12 Statistical distribution tables 80 12.1 Normal distribution table . 80 12.2 Chi-square distribution table . 81 12.3 Tails of t-distribution . 82 12.4 Critical values of the F-distribution . 83 12.5 Critical values of the Studentised range distribution . 86 12.6 Critical values for the rank sum test . 88 2 12.7 Critical values for the signed rank test . 89 13 Solutions to exercises 91 13.1 Solutions to Section 2 (random sampling) . ..